The specific heat of the quasi-one-dimensional organic compound (TMTSF)(2)ClO4 was measured between 1.9 and 35 K for two stares corresponding to the metallic/superconducting ground state and the insulating spin-density-wave (SDW) ground state at low temperature, by varying the cooling conditions through the anion-ordering (AO) transition at 24 K. These are the first thermodynamical measurements of the SDW ground state. Like in the previously investigated compounds (TMTSF)(2)X, with X = PF6 and AsF6, and (TMTTF)(2)Br (where TMTTF equivalent to tetramethyltetrathiafulvalene), the lattice specific heat shows strong deviations from a Debye behaviour, which can be explained by the quasi-one-dimensional structural character Bumps in C/T-3 occur at T = 3.5-4 K and 5 K for the quenched SDW and relaxed metallic states respectively; these are in good agreement with the frequencies of low-tying modes detected by infrared spectroscopy for the metallic sample. In addition, analysis of the lattice contribution suggests a T-alpha (alpha = 2.7) acoustic contribution for T greater than or similar to 2 K, ascribed to the low dimensionality of the structure. The SDW transition was detected at T-C = 4.5 K, as expected from the high value of the quenching rate (of 3.3 K s(-1)) through the AO transition (the amplitude of the anomaly represents about 3% of the total heat capacity). The AO transition was investigated either by decreasing the temperature for the slow-cooled sample or by reheating, at a similar rate, for the quenched sample; the specific heat anomalies are identical for the two procedures, due to reordering annealing effects on reheating, a phenomenon previously investigated by x-ray experiments. Surprisingly, the entropy under the AO anomaly is R ln (4/3), much lower than the R ln 2 value expected for the two possible configurational orientations of the perchlorate ion in the disordered state. Similar low values of the entropy have been previously measured for the perrhenate (ReO4) and BF4 salts.